Filip Kostic scored in both halves in Barcelona as Frankfurt reached the semi-finals
After the first leg had finished 1-1, the German side stormed into a 3-0 lead in the second and looked in control before two late goals for Barcelona.
Filip Kostic's penalty and Rafael Borre's superb strike were followed by another for Kostic after the break.
But Sergio Busquets' effort and Memphis Depay's penalty set up a tense finish.
Frankfurt defender Evan Ndicka was also sent off for a second yellow card for what looked a soft foul on Luuk de Jong for the spot-kick, with nine minutes added on following a VAR malfunction.
But Frankfurt, who sit ninth in the Bundesliga and won this competition in its previous guise in 1980, saw out a memorable victory to the delight of their almost 30,000 travelling fans.
Barcelona, on a superb run since the arrival of Xavi, did not know how to cope with the Bundesliga outfit's clinical counter-attacks and Frankfurt could have had more.
Kostic slotted the first from the spot following a foul by Eric Garcia but the goal of the game came from Borre, who thundered a strike into the roof of the net from distance.
The hosts had chances of their own and Pierre-Emerick Aubameyang headed over in the first half before failing to connect with Ousmane Dembele's low cross in the second.
Busquets saw an initial effort ruled out for offside in added time before sparking a dramatic finish with a well-taken goal, but the hosts ran out of time after Depay's penalty just crept over the line.
It means Frankfurt can look forward to a semi-final against David Moyes' Hammers.
-- Courtesy of BBC Sport